2025/11/12 by Πέτρος Γαλανόπουλος, Galanopoulos, Petros, Daniel Girela +1
Mathematics · #30H10 #42B30 #47B91 #Advanced Banach Space Theory #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2511.09201
openalex publication_date 2025/11/12 · openalex created_date 2025/11/14 · openalex updated_date 2026/07/31
In this article we address the question of characterizing the sequences of complex numbers (η)=\ ηn\n=0^∞ whose associated Rhaly operator \mathcal R(η) is bounded or compact on the Hardy spaces Hp (1≤ p<∞ ), on the Bergman spaces Apα, and on the Dirichlet spaces \mathcal Dpα (1≤ p<∞ , α>-1). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of \mathcal R(η) on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function F(η) defined by F(η)(z)=∑n=0^∞ ηnzn (z∈ \mathbb D). \par We prove that if 2≤ p<∞ and ηn=\og ((1)/(n) ), then \mathcal R(η) is bounded on Hp. However, there exists a sequence (η) with ηn=\og ((1)/(n) ) such that the operator \mathcal R(η) is not bounded on Hp for 1≤ p<2. \par We deal also with the derivative-Hardy spaces. For p>0 the derivative-Hardy space Sp consists of those functions f, analytic in the unit disc \mathbb D, such that f^′ ∈ Hp. We prove that if 1≤ p<∞ and 1